Skip to main content

Full text of "Men Of Mathematics"

See other formats

on a difficult point which Gauss had left aside concerning the
solution hi integers x, y, z of the indeterminate equation
oz2 + faf -r cz* = 0,
where a, b, c are any given integers. This was a fine piece of
work, but it is safe to say that no mathematician who read it
anticipated that the conservative author of twenty-two was to
become one of the most radical originators in the history of
mathematics. Talent no doubt is plain enough in this first
attempts but genius - no. There is not a single hint of the great
originator hi this severely classical dissertation.
The like may be said for all of Cantor's earliest work pub-
lished before he was twenty-nine. It was excellent, but might
have been done by any brilliant man who had thoroughly
absorbed, as Cantor had, the doctrine of rigorous proof from
Gauss and Weierstrass. Cantor's first love was the Gaussian
theory of numbers, to which he was attracted by the hard,
sharp, clear perfection of the proofs. From this, under the influ-
ence of the Weierstrassians, he presently branched off into
rigorous analysis, particularly in the theory of trigonometric
series (Fourier series).
The subtle difficulties of this theory (where questions of con-
vergence of infinite series are less easily approachable than in
the theory of power series) seem to have inspired Cantor to go
deeper for the foundations of analysis than any of his contem-
poraries had cared to look, and he was led to his grand attack
on the mathematics and philosophy of the infinite itself, which
is at the bottom of all questions concerning continuity, limits,
and convergence. Just before he was thirty, Cantor published
his first revolutionary paper (hi Crelle's Journal) on the theory
of infinite sets. This will be described presently. The unex-
pected and paradoxical result concerning the set of all algebraic
numbers which Cantor established in this paper and the com-
plete novelty of the methods employed immediately marked the
young author as a creative mathematician of extraordiiiary
originalitj-. Whether all agreed that the new methods were
sound or not is beside the point; it was universally admitted
that a man had arrived with something fundamentally new hi